Vertical and horizontal asymptotes practice problems
Vertical asymptotes from the zeros of the denominator, horizontal ones from the end behaviour. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{3 x^{2} - 9 x - 30}{x^{2} - 5 x + 4} \).
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{- 3 x^{2} - 12 x - 12}{x^{2} - 16} \).
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{8 - 4 x}{x^{2} - 5 x + 4} \).
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{2 x + 8}{x^{2} - 4 x + 3} \).
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{x^{2} + 3 x + 2}{x^{2} + x - 12} \).
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{- 4 x - 16}{x^{2} - 1} \).
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{- 4 x^{3} + 12 x^{2} - 4 x + 12}{x^{2} + 2 x - 8} \).
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{- 4 x^{2} + 28 x - 48}{x^{2} + 5 x + 4} \).
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{4 x^{3} - 4 x^{2} + 4 x - 4}{x^{2} - x - 2} \).
Find all vertical and horizontal asymptotes of \( \displaystyle f(x) = \frac{- x^{2} + 9 x - 20}{x^{2} + 7 x + 12} \).